Integrand size = 22, antiderivative size = 92 \[ \int \frac {\sqrt {x} \left (A+B x^n\right )}{\left (a+b x^n\right )^2} \, dx=\frac {(A b-a B) x^{3/2}}{a b n \left (a+b x^n\right )}+\frac {(3 a B-A b (3-2 n)) x^{3/2} \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 n},1+\frac {3}{2 n},-\frac {b x^n}{a}\right )}{3 a^2 b n} \] Output:
(A*b-B*a)*x^(3/2)/a/b/n/(a+b*x^n)+1/3*(3*B*a-A*b*(3-2*n))*x^(3/2)*hypergeo m([1, 3/2/n],[1+3/2/n],-b*x^n/a)/a^2/b/n
Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {x} \left (A+B x^n\right )}{\left (a+b x^n\right )^2} \, dx=\frac {x^{3/2} \left (-\frac {3 a (-A b+a B)}{a+b x^n}+(3 a B+A b (-3+2 n)) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 n},1+\frac {3}{2 n},-\frac {b x^n}{a}\right )\right )}{3 a^2 b n} \] Input:
Integrate[(Sqrt[x]*(A + B*x^n))/(a + b*x^n)^2,x]
Output:
(x^(3/2)*((-3*a*(-(A*b) + a*B))/(a + b*x^n) + (3*a*B + A*b*(-3 + 2*n))*Hyp ergeometric2F1[1, 3/(2*n), 1 + 3/(2*n), -((b*x^n)/a)]))/(3*a^2*b*n)
Time = 0.36 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {957, 888}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {x} \left (A+B x^n\right )}{\left (a+b x^n\right )^2} \, dx\) |
\(\Big \downarrow \) 957 |
\(\displaystyle \frac {(3 a B-A b (3-2 n)) \int \frac {\sqrt {x}}{b x^n+a}dx}{2 a b n}+\frac {x^{3/2} (A b-a B)}{a b n \left (a+b x^n\right )}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {x^{3/2} (3 a B-A b (3-2 n)) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 n},1+\frac {3}{2 n},-\frac {b x^n}{a}\right )}{3 a^2 b n}+\frac {x^{3/2} (A b-a B)}{a b n \left (a+b x^n\right )}\) |
Input:
Int[(Sqrt[x]*(A + B*x^n))/(a + b*x^n)^2,x]
Output:
((A*b - a*B)*x^(3/2))/(a*b*n*(a + b*x^n)) + ((3*a*B - A*b*(3 - 2*n))*x^(3/ 2)*Hypergeometric2F1[1, 3/(2*n), 1 + 3/(2*n), -((b*x^n)/a)])/(3*a^2*b*n)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a *b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* (p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N eQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 , m, (-n)*(p + 1)]))
\[\int \frac {\sqrt {x}\, \left (A +B \,x^{n}\right )}{\left (a +b \,x^{n}\right )^{2}}d x\]
Input:
int(x^(1/2)*(A+B*x^n)/(a+b*x^n)^2,x)
Output:
int(x^(1/2)*(A+B*x^n)/(a+b*x^n)^2,x)
\[ \int \frac {\sqrt {x} \left (A+B x^n\right )}{\left (a+b x^n\right )^2} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \sqrt {x}}{{\left (b x^{n} + a\right )}^{2}} \,d x } \] Input:
integrate(x^(1/2)*(A+B*x^n)/(a+b*x^n)^2,x, algorithm="fricas")
Output:
integral((B*x^n + A)*sqrt(x)/(b^2*x^(2*n) + 2*a*b*x^n + a^2), x)
Result contains complex when optimal does not.
Time = 23.77 (sec) , antiderivative size = 932, normalized size of antiderivative = 10.13 \[ \int \frac {\sqrt {x} \left (A+B x^n\right )}{\left (a+b x^n\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(x**(1/2)*(A+B*x**n)/(a+b*x**n)**2,x)
Output:
A*(6*a*a**(3/(2*n))*a**(-2 - 3/(2*n))*n*x**(3/2)*lerchphi(b*x**n*exp_polar (I*pi)/a, 1, 3/(2*n))*gamma(3/(2*n))/(4*a*n**3*gamma(1 + 3/(2*n)) + 4*b*n* *3*x**n*gamma(1 + 3/(2*n))) + 6*a*a**(3/(2*n))*a**(-2 - 3/(2*n))*n*x**(3/2 )*gamma(3/(2*n))/(4*a*n**3*gamma(1 + 3/(2*n)) + 4*b*n**3*x**n*gamma(1 + 3/ (2*n))) - 9*a*a**(3/(2*n))*a**(-2 - 3/(2*n))*x**(3/2)*lerchphi(b*x**n*exp_ polar(I*pi)/a, 1, 3/(2*n))*gamma(3/(2*n))/(4*a*n**3*gamma(1 + 3/(2*n)) + 4 *b*n**3*x**n*gamma(1 + 3/(2*n))) + 6*a**(3/(2*n))*a**(-2 - 3/(2*n))*b*n*x* *(3/2)*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 3/(2*n))*gamma(3/(2*n))/ (4*a*n**3*gamma(1 + 3/(2*n)) + 4*b*n**3*x**n*gamma(1 + 3/(2*n))) - 9*a**(3 /(2*n))*a**(-2 - 3/(2*n))*b*x**(3/2)*x**n*lerchphi(b*x**n*exp_polar(I*pi)/ a, 1, 3/(2*n))*gamma(3/(2*n))/(4*a*n**3*gamma(1 + 3/(2*n)) + 4*b*n**3*x**n *gamma(1 + 3/(2*n)))) + B*(4*a*a**(-3 - 3/(2*n))*a**(1 + 3/(2*n))*n**2*x** (n + 3/2)*gamma(1 + 3/(2*n))/(4*a*n**3*gamma(2 + 3/(2*n)) + 4*b*n**3*x**n* gamma(2 + 3/(2*n))) - 6*a*a**(-3 - 3/(2*n))*a**(1 + 3/(2*n))*n*x**(n + 3/2 )*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1 + 3/(2*n))*gamma(1 + 3/(2*n))/(4 *a*n**3*gamma(2 + 3/(2*n)) + 4*b*n**3*x**n*gamma(2 + 3/(2*n))) + 6*a*a**(- 3 - 3/(2*n))*a**(1 + 3/(2*n))*n*x**(n + 3/2)*gamma(1 + 3/(2*n))/(4*a*n**3* gamma(2 + 3/(2*n)) + 4*b*n**3*x**n*gamma(2 + 3/(2*n))) - 9*a*a**(-3 - 3/(2 *n))*a**(1 + 3/(2*n))*x**(n + 3/2)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1 + 3/(2*n))*gamma(1 + 3/(2*n))/(4*a*n**3*gamma(2 + 3/(2*n)) + 4*b*n**3*...
\[ \int \frac {\sqrt {x} \left (A+B x^n\right )}{\left (a+b x^n\right )^2} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \sqrt {x}}{{\left (b x^{n} + a\right )}^{2}} \,d x } \] Input:
integrate(x^(1/2)*(A+B*x^n)/(a+b*x^n)^2,x, algorithm="maxima")
Output:
4*((2*n^2 - 3*n)*A*a*b + 3*B*a^2*n)*integrate(sqrt(x)/((8*n^2 - 18*n + 9)* b^4*x^(3*n) + 3*(8*n^2 - 18*n + 9)*a*b^3*x^(2*n) + 3*(8*n^2 - 18*n + 9)*a^ 2*b^2*x^n + (8*n^2 - 18*n + 9)*a^3*b), x) - 2*(B*b*(4*n - 3)*x*x^n + (A*b* (2*n - 3) + 4*B*a*n)*x)*sqrt(x)/((8*n^2 - 18*n + 9)*b^3*x^(2*n) + 2*(8*n^2 - 18*n + 9)*a*b^2*x^n + (8*n^2 - 18*n + 9)*a^2*b)
\[ \int \frac {\sqrt {x} \left (A+B x^n\right )}{\left (a+b x^n\right )^2} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \sqrt {x}}{{\left (b x^{n} + a\right )}^{2}} \,d x } \] Input:
integrate(x^(1/2)*(A+B*x^n)/(a+b*x^n)^2,x, algorithm="giac")
Output:
integrate((B*x^n + A)*sqrt(x)/(b*x^n + a)^2, x)
Timed out. \[ \int \frac {\sqrt {x} \left (A+B x^n\right )}{\left (a+b x^n\right )^2} \, dx=\int \frac {\sqrt {x}\,\left (A+B\,x^n\right )}{{\left (a+b\,x^n\right )}^2} \,d x \] Input:
int((x^(1/2)*(A + B*x^n))/(a + b*x^n)^2,x)
Output:
int((x^(1/2)*(A + B*x^n))/(a + b*x^n)^2, x)
\[ \int \frac {\sqrt {x} \left (A+B x^n\right )}{\left (a+b x^n\right )^2} \, dx=\int \frac {\sqrt {x}}{x^{n} b +a}d x \] Input:
int(x^(1/2)*(A+B*x^n)/(a+b*x^n)^2,x)
Output:
int(sqrt(x)/(x**n*b + a),x)